Lemma 27.2.5.label Let $G$ be a locally compact group, $f, f'\in C_{c}^{+}(G)$, and $\eps > 0$, then there exists $V \in \cn_{G}(1)$ such that for any $g \in C_{c}^{+}(V)$ with $g \ne 0$,
Proof, [Lemma 2.18, Fol16]. By Urysohn’s Lemma, there exists $\eta \in C_{c}^{+}(G; [0, 1])$ such that $\eta|_{\supp{f} \cup \supp{f'}}= 1$.
Let $\delta > 0$, and define
By Proposition 27.1.2, there exists $V \in \cn_{G}(1)$ such that for any $x, y \in G$ with $x^{-1}y \in V$,
Let $g \in C_{c}^{+}(V)$, $\seqf{c_j}\subset [0, \infty)$, and $\seqf{x_j}\subset G$ such that $H \le \sum_{j = 1}^{n} c_{j} L_{x_j}\phi$, then for each $x \in G$,
Likewise,
As $h + h' \le 1$,
Since the above holds for all such $\seqf{c_j}\subset [0, \infty)$ and $\seqf{x_j}\subset G$,
$\square$
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